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Probabilistic Design and Reliability | Module 7 of 10

Uncertainty Propagation and Sensitivity

Probabilistic design needs more than an output histogram. It needs to know which uncertain inputs control the result and which impressive-looking variables barely matter.

Readiness check

Tick only what you can do closed-notes before using this module for design decisions.

Take or interpret a partial derivative.
Run a parameter sweep in code.
Read an output distribution summary.
Explain correlation in plain language.
Link a sensitivity result to a design decision.

0 or 1 weak itemsContinue with this module.

2 weak itemsReview uncertainty propagation in VVUQ, Module 7.

3 or more weak itemsReview numerical sensitivity in VVUQ, Module 8.

The core idea

Uncertainty propagation predicts output scatter. Sensitivity analysis explains which inputs cause that scatter and which design or data improvements are worth attention.

local sensitivity: S_i = partial y / partial x_ilinear variance: var(y) approx sum(S_i^2 var(x_i))rank effect: change input, measure output change

This course uses uncertainty propagation for design reliability, while the VVUQ course goes deeper into validation and model credibility. The workflow here is practical: define uncertain inputs, compute output distribution, identify influential variables, then choose a design or data action. Local sensitivity uses derivatives near a nominal point, which is efficient but can miss nonlinear effects. Sampling-based sensitivity changes one input or groups of inputs across a distribution and watches output variation. Correlation matters because inputs can move together. A diameter tolerance and a surface finish process may share manufacturing causes. A load and temperature may rise together in service. Sensitivity helps avoid spending effort on variables that are visually dramatic but reliability-irrelevant.

The skill works when: the failure event, units, assumptions, distributions, and checks are visible.

The skill breaks down when: a probability is reported without the mechanical model and evidence boundary.

The concept. Uncertainty propagation predicts output scatter. Sensitivity analysis explains which inputs cause that scatter and which design or data improvements are worth attention.

The skills, taught in order

Propagate before redesign

Measure output scatter before deciding which variable to improve.

Use local sensitivity for quick screening

Derivatives show which inputs matter near a nominal design.

Use sampling when nonlinear

Sampling reveals skew, nonlinear thresholds, and tail behavior.

Rank influence by contribution

A variable matters when its uncertainty and sensitivity combine to move the output.

Test correlation effects

If dependence is plausible, rerun the analysis with correlation scenarios.

Engineering connection: use these skills to turn variability into a design decision, not just a plot.

Worked example 1: diameter sensitivity in shaft stress

Torsional stress in a solid shaft is tau = 16T/(pi d^3). At T = 400000 N mm and d = 28 mm, compare relative sensitivity to torque and diameter.

ProblemTorsional stress in a solid shaft is tau = 16T/(pi d^3). At T = 400000 N mm and d = 28 mm, compare relative sensitivity to torque and diameter.
AssumptionsThe equation is valid and inputs vary slightly near nominal values.
Modeld ln(tau) / d ln(T) = 1d ln(tau) / d ln(d) = -3
SolveA 1 percent increase in torque increases stress by 1 percent. A 1 percent increase in diameter decreases stress by 3 percent. Diameter variation has three times the relative leverage of torque variation per percent change.
CheckBecause d is cubed in the denominator, small diameter changes strongly affect stress.
ConclusionDiameter tolerance can dominate stress uncertainty even when torque uncertainty looks more obvious.

Result. Diameter tolerance can dominate stress uncertainty even when torque uncertainty looks more obvious.

Worked example 2: correlation effect on output variance

A simplified load effect is Y = X1 + X2. Both inputs have standard deviation 10 units. Compare output standard deviation for rho = 0 and rho = 0.8.

ProblemA simplified load effect is Y = X1 + X2. Both inputs have standard deviation 10 units. Compare output standard deviation for rho = 0 and rho = 0.8.
AssumptionsInputs have equal variance and linear correlation.
Modelvar(Y) = sigma_1^2 + sigma_2^2 + 2 rho sigma_1 sigma_2
SolveFor rho = 0, sigma_Y = sqrt(100 + 100) = 14.1. For rho = 0.8, sigma_Y = sqrt(100 + 100 + 160) = 19.0.
CheckPositive correlation increases high-output tails because inputs move upward together.
ConclusionIgnoring positive correlation can understate output scatter by about 34 percent in this example.

Result. Ignoring positive correlation can understate output scatter by about 34 percent in this example.

Python activity

Engineering question: Rank uncertain inputs for a shaft stress model by standardized perturbation.

Assumptions and units: Use the units shown in the code comments. Keep the random seed fixed while learning so the result is reproducible.

from math import pi

def tau(torque, diameter):
    return 16.0 * torque / (pi * diameter**3)

t0 = 400_000.0
d0 = 28.0
base = tau(t0, d0)

for name, t, d in [
    ("torque +1 percent", 1.01 * t0, d0),
    ("diameter +1 percent", t0, 1.01 * d0),
    ("diameter -1 percent", t0, 0.99 * d0),
]:
    change = (tau(t, d) - base) / base
    print(name, "relative stress change", round(change, 4))

Numerical checks: Diameter changes should cause about three times the relative stress response of torque changes, with opposite sign for increasing diameter.

Limitations: This local perturbation does not replace full distribution sampling when tolerances are large or nonlinear thresholds exist.

Misconceptions and diagnostics

Mistake	Diagnostic question	Correction
The model output distribution is trustworthy because the code executed.	Ask: were units, assumptions, convergence, and benchmark cases checked?	Correct the assumption, then rerun the calculation or rewrite the report.
A colorful sensitivity plot proves causal importance.	Ask: what uncertainty range and model equation produced the ranking?	Correct the assumption, then rerun the calculation or rewrite the report.
The variable with the biggest unit range must matter most.	Ask: what is the output sensitivity and standardized variation?	Correct the assumption, then rerun the calculation or rewrite the report.
Correlation can be ignored if it is inconvenient.	Ask: what shared cause might make inputs move together?	Correct the assumption, then rerun the calculation or rewrite the report.

Practice ladder

Recognize

Name one input in a shaft model likely to have high sensitivity because of an exponent.

Solution guidance

Write the governing equation, state units and assumptions, then compare the answer with the relevant limit state or design decision.

Calculate

For Y = 3X, if sigma_X = 2, find sigma_Y.

Solution guidance

Write the governing equation, state units and assumptions, then compare the answer with the relevant limit state or design decision.

Diagnose

A sensitivity study ranks force in N above diameter in m because the force numbers are larger. Identify the scaling error.

Solution guidance

Write the governing equation, state units and assumptions, then compare the answer with the relevant limit state or design decision.

Design

Plan a sensitivity check that would decide whether to tighten a diameter tolerance or improve load measurement.

Solution guidance

Write the governing equation, state units and assumptions, then compare the answer with the relevant limit state or design decision.

Working with AI, and proving it yourself

Useful AI support

Ask for alternative explanations of terminology, candidate code tests, or a checklist of assumptions to verify.

Ask it to challenge a derivation, then prove the result with units, limiting cases, and an independent calculation.

Do not let AI invent material distributions, fabricate failure data, claim standards compliance, or make final safety decisions.

Portfolio evidence

Create one short reliability note for this module. It must include the engineering question, assumptions, input definitions, equations, calculation, checks, interpretation, and limitations.

Verification required: dimensional analysis, a benchmark case, convergence or sample-size check, and a plain-language residual-risk statement.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

What does propagation answer?

How uncertain inputs create uncertain outputs.

What does sensitivity answer?

Which inputs drive the output variation or failure probability.

When is local sensitivity useful?

Near a nominal point for smooth, small changes.

Why test correlation scenarios?

Dependence can change output scatter and tail probability.

How does this differ from VVUQ?

Here the focus is probabilistic design decisions; VVUQ focuses more deeply on model credibility and validation.

TodayFinish the review and first two practice levels.

+1 dayRebuild one worked example from a blank page.

+3 daysChange one assumption and predict the effect.

+7 daysConnect this module to the capstone bracket.

+30 daysReuse the method in a new mechanical component.

References and source discipline

This course synthesizes original MechCompass explanations from established reliability engineering sources. Confirm current editions and standards before using them for formal design approval.

Topic in this module	Reference direction
Sensitivity in reliability analysis	Haldar and Mahadevan, Probability, Reliability, and Statistical Methods in Engineering Design
Engineering uncertainty propagation	NIST/SEMATECH Engineering Statistics Handbook
VVUQ relationship and uncertainty classification	ASME V&V standards, confirm current edition before formal use